Viscous Liquid-gas Two-phase Flow Model Research Articles

Global strong solutions to the viscous liquid–gas two-phase flow model with slip boundary conditions in 3D exterior domains

This paper is concerned with the viscous liquid–gas two-phase flow model in a three-dimensional exterior domain with the slip boundary conditions. We are able to prove the existence of a global strong solution when the initial total energy is suitably small. The initial masses of liquid and gas are allowed to contain a vacuum, with no extra restriction between the initial masses of liquid and gas. It should be noted here that we consider the slip boundary condition in a three dimensional (3D) exterior domain which is in sharp contrast to result of Yu (2021) where they consider the Cauchy problem.

Global well-posedness and large time behaviour of the viscous liquid-gas two-phase flow model in ℝ3

AbstractWe investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

Review on mathematical analysis of some two-phase flow models

The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.

Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain

In this paper, we investigate global existence and asymptotic behavior of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain with no-slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in $H^2(Ω)$ . Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase flow model in \begin{document}$\mathbb{R}^N\,(N≥2)$ \end{document} . We concentrate on the critical Besov spaces based on the \begin{document}$L^p$ \end{document} setting. We improve the range of Lebesgue exponent \begin{document}$p$ \end{document} , for which the system is locally well-posed, compared to [ 22 ]. Applying Lagrangian coordinates is the key to our statements, as it enables us to obtain the result by means of Banach fixed point theorem.

Large time behavior for the system of a viscous liquid–gas two-phase flow model in [formula omitted

The Cauchy problem of a three-dimensional compressible viscous liquid–gas two-phase flow model is considered in the present paper. The global existence and uniqueness of solutions are established when the initial data is close to its equilibrium in the framework of Sobolev space H3(R3). Moreover, the optimal L2–L2 convergence rates are also obtained for the solution.

Decay of the 3D viscous liquid-gas two-phase flow model with damping

We establish the optimal Lp − L2(1 ≤ p < 6/5) time decay rates of the solution to the Cauchy problem for the 3D viscous liquid-gas two-phase flow model with damping and analyse the influences of the damping on the qualitative behaviors of solution. It is observed that the fraction effect of the damping affects the dispersion of fluids and enhances the time decay rate of solution. Our method of proof consists of Hodge decomposition technique, Lp − L2 estimates for the linearized equations, and delicate energy estimates.

A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

In this paper, we establish a new blowup criterion for the strongsolutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In[13], Wen, Yao, and Zhu prove that if the strongsolutions blow up at finite time $T^*$, the mass in$L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extendWen, Yao, and Zhu's work in the sense of the concentration of massin $BMO(\Omega)$ norm at $T^*$. The method can be applied to studythe blow-up criterion in terms of the concentration of density in$BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in[11]. Moreover, the appearance of vacuum is allowedin the paper.

Decay of the 3D inviscid liquid–gas two-phase flow model

We establish the optimal \({L^{p}-L^{2}(1 \leq p < 6/5)}\) time decay rates of the solution to the Cauchy problem for the 3D inviscid liquid–gas two-phase flow model and analyze the influences of the damping on the qualitative behaviors of solution. Compared with the viscous liquid–gas two-phase flow model (Zhang and Zhu in J Differ Equ 258:2315–2338, 2015), our results imply that the friction effect of the damping is stronger than the dissipation effect of the viscosities and enhances the decay rate of the velocity. Our proof is based on Hodge decomposition technique, the \({L^{p}-L^{2}}\) estimates for the linearized equations and an elaborate energy method.

Blow up criterion of strong solution for 3D viscous liquid–gas two-phase flow model with vacuum

In this paper, we establish a blow-up criterion to the local strong solution to the three dimensional (3D) viscous liquid–gas two-phase flow model only in terms of the divergence of the velocity field. Moreover, the initial vacuum is allowed, and there is no extra restriction on viscous coefficients. Both the Cauchy problem and initial-boundary value problem are considered in this paper.

On the well-posedness for a multi-dimensional compressible viscous liquid–gas two-phase flow model in critical spaces

This paper is dedicated to study of the Cauchy problem for a multi-dimensional (\({N \geq 2}\)) compressible viscous liquid–gas two-phase flow model. We prove the local well-posedness of the system for large data in critical Besov spaces based on the L p framework under the sole assumption that the initial liquid mass is bounded away from zero.

Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our ideamainly comes from [23] and [29] which describe anisothermal Navier-Stokes equation in a half line. We obtain theconvergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. These theorems are proved by a weighted energy method.

Global existence and optimal convergence rates for the strong solutions in [formula omitted] to the 3D viscous liquid–gas two-phase flow model

This paper is concerned with the Cauchy problem of the three-dimensional viscous liquid–gas two-phase flow model. We prove the global existence of a strong solution when H2-norm of the initial perturbation around a constant state is sufficiently small and its L1-norm is bounded. Moreover, the optimal convergence rates are also obtained for such a solution.

A blow-up criterion of strong solutions to a viscous liquid–gas two-phase flow model with vacuum in 3D

In this paper, we get a blow-up criterion for the strong solutions to the 3D viscous liquid–gas two-phase flow model with vacuum. More precisely, we prove that the bound of Lt1Lx∞ norm of the deformation tensor of velocity gradient controls the possible breakdown of the strong solutions.

Blow-up criterion for 3D viscous liquid–gas two-phase flow model

This work deals with the blow-up criterion for the strong solution to the three-dimensional viscous liquid–gas two-phase flow model in terms of the L1(0,T;L∞)-norm of the gradient of the velocity with two types of boundary conditions: the Dirichlet boundary condition and the Navier-slip boundary condition. There is no extra restriction on viscosity coefficients. The result also applies to the whole space case.

A note on viscous liquid–gas two-phase flow model with mass-dependent viscosity and vacuum

In this paper, we consider a free boundary value problem for two-phase liquid–gas model with mass-dependent viscosity coefficient; the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid, and the fluid velocities are unequal, i.e., ug≠ul. The local existence of a weak solution is established when the initial gas mass connects to vacuum continuously.

Incompressible Limit of Viscous Liquid-Gas Two-Phase Flow Model

We investigate the incompressible limit of the viscous liquid-gas two-phase flow model with periodic boundary conditions. It is shown that the local classical solution of the two-phase flow model converges to the local classical solution of the incompressible Navier--Stokes equations, as $\epsilon\rightarrow \infty$, which builds up the relationship between the two-phase flow model and the incompressible Navier--Stokes equations. In addition, we give the convergence rate estimates in some norms.

Well-Posedness for a Multidimensional Viscous Liquid-Gas Two-Phase Flow Model

The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.

A blow-up criterion of strong solution to a 3D viscous liquid–gas two-phase flow model with vacuum

In this paper, we get a unique local strong solution to a 3D viscous liquid–gas two-phase flow model in a smooth bounded domain. Besides, a blow-up criterion of the strong solution for 253μ>λ is obtained. The method can be applied to study a blow-up criterion of the strong solution to Navier–Stokes equations for 253μ>λ, which improves the corresponding result about Navier–Stokes equations in Sun et al. (2011) [15] where 7μ>λ. Moreover, all the results permit the appearance of vacuum.

Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum

This paper is concerned with the Cauchy problem of the 3D viscous liquid-gas two-phase flow model, where the initial vacuum is allowed. The existence of the global strong solution is proved when the energy of the initial data is small enough.